Exponential Growth

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Answer Key & Explanations

Exponential Growth · Grade 9 · Worksheet 2

1. A research lab is studying bacterial growth in a petri dish. The initial population is 200 bacteria, and the population doubles every 3 hours. Write an exponential function in the form P(t) = P₀ * a^t that models the bacterial population after t hours, where P₀ is the initial population and a is the growth factor per hour. Answer: P(t) = 200 * 2^(t/3) Solution: Initial population \( P_0 = 200 \) bacteria. The population doubles every 3 hours. P(t) = P_0 \cdot a^t where \( t \) is in hours, \( a \) is the hourly growth factor.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the given information** Initial population \( P_0 = 200 \) bacteria. The population doubles every 3 hours. We want an exponential function of the form \[ P(t) = P_0 \cdot a^t \] where \( t \) is in hours, \( a \) is the hourly growth factor. --- **Step 2: Understand what "doubles every 3 hours" means** If \( t = 3 \) hours, then \( P(3) = 2 \times P_0 \). So: \[ 200 \cdot a^3 = 200 \times 2 \] Divide both sides by 200: \[ a^3 = 2 \] --- **Step 3: Solve for \( a \)** \[ a = 2^{1/3} \] So the hourly growth factor is \( 2^{1/3} \). --- **Step 4: Write the function with this \( a \)** \[ P(t) = 200 \cdot (2^{1/3})^t \] Simplify the exponent: \[ P(t) = 200 \cdot 2^{t/3} \] --- **Step 5: Verify the doubling behavior** Check \( t = 3 \): \[ P(3) = 200 \cdot 2^{3/3} = 200 \cdot 2^1 = 400 \] That’s double the initial population, correct. --- **Final Answer:** \[ P(t) = 200 \cdot 2^{t/3} \]

2. Hana is comparing the growth of two functions on a coordinate plane: the exponential function f(x) = 12^x and the polynomial function g(x) = x^12. She graphs both for x ≥ 1 and observes that the polynomial function is larger for some small values of x. Determine the smallest integer value of x greater than 1 for which f(x) = 12^x first exceeds g(x) = x^12. Answer: 13 Solution: Evaluate f(2) = 12^2 = 144, g(2) = 2^12 = 4096. f(2) < g(2). Evaluate f(3) = 12^3 = 1728, g(3) = 3^12 = 531441.
   Full step-by-step solution

   Step 1: Evaluate f(2) = 12^2 = 144, g(2) = 2^12 = 4096. f(2) < g(2). Step 2: Evaluate f(3) = 12^3 = 1728, g(3) = 3^12 = 531441. f(3) < g(3). Step 3: Evaluate f(4) = 12^4 = 20736, g(4) = 4^12 = 16777216. f(4) < g(4). Step 4: Evaluate f(5) = 12^5 = 248832, g(5) = 5^12 = 244140625. f(5) < g(5). Step 5: Evaluate f(6) = 12^6 = 2985984, g(6) = 6^12 = 2176782336. f(6) < g(6). Step 6: Evaluate f(7) = 12^7 = 35831808, g(7) = 7^12 = 13841287201. f(7) < g(7). Step 7: Evaluate f(8) = 12^8 = 429981696, g(8) = 8^12 = 68719476736. f(8) < g(8). Step 8: Evaluate f(9) = 12^9 = 5159780352, g(9) = 9^12 = 282429536481. f(9) < g(9). Step 9: Evaluate f(10) = 12^10 = 61917364224, g(10) = 10^12 = 1000000000000. f(10) < g(10). Step 10: Evaluate f(11) = 12^11 = 743008370688, g(11) = 11^12 = 3138428376721. f(11) < g(11). Step 11: Evaluate f(12) = 12^12 = 8916100448256, g(12) = 12^12 = 8916100448256. f(12) = g(12). Step 12: Evaluate f(13) = 12^13 = 106993205379072, g(13) = 13^12 = 23298085122481. f(13) > g(13). Thus, the smallest integer x greater than 1 where f(x) exceeds g(x) is x = 13. The answer is 13.

3. A rare orchid species in a botanical garden is growing exponentially. The number of orchids is modeled by the function O(t) = 120 × 2^(t/4), where t is the time in months. How many orchids will there be after 12 months? Answer: 960 Solution: The function is O(t) = 120 × 2^(t/4) Substitute t = 12 into the function: O(12) = 120 × 2^(12/4) Simplify the exponent: 12/4 = 3 Now we have O(12) = 120 × 2^3 Calculate 2^3 = 8 Multiply: 120 × 8 = 960 The answer is 960 orchids.
   Full step-by-step solution

   Step 1: The function is O(t) = 120 × 2^(t/4) Step 2: Substitute t = 12 into the function: O(12) = 120 × 2^(12/4) Step 3: Simplify the exponent: 12/4 = 3 Step 4: Now we have O(12) = 120 × 2^3 Step 5: Calculate 2^3 = 8 Step 6: Multiply: 120 × 8 = 960 Step 7: The answer is 960 orchids.

4. A colony of bacteria doubles in size every 3 hours. If the colony starts with 500 bacteria, how many bacteria will there be after 15 hours? Answer: 16000 Solution: The colony starts with 500 bacteria. It doubles every 3 hours. We want the number after 15 hours.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** The colony starts with 500 bacteria. It doubles every 3 hours. We want the number after 15 hours. --- **Step 2: Determine how many doubling periods occur in 15 hours** Doubling time = 3 hours. Number of periods = Total time / Doubling time = 15 / 3 = 5 periods. So the colony doubles 5 times. --- **Step 3: Set up the growth formula** Initial amount = 500 After 1 doubling: 500 × 2 After 2 doublings: 500 × 2 × 2 = 500 × 2^2 After n doublings: 500 × 2^n Here n = 5. Final amount = 500 × 2^5 --- **Step 4: Calculate 2^5** 2^5 = 2 × 2 × 2 × 2 × 2 = 32 --- **Step 5: Multiply by initial population** Final amount = 500 × 32 500 × 30 = 15,000 500 × 2 = 1,000 Total = 15,000 + 1,000 = 16,000 --- **Step 6: Conclusion** After 15 hours, there will be 16,000 bacteria. --- **Final answer: 16000**

5. On a coordinate plane, Isabella graphs two functions: f(x) = 3^x and g(x) = x^3. At x = 2, f(2) = 9 and g(2) = 8, so f(x) is slightly larger. At x = 3, f(3) = 27 and g(3) = 27, so they are equal. For what positive integer value of x greater than 3 does f(x) first exceed g(x) by more than 1000? Answer: 7 Solution: Calculate f(4) = 3^4 = 81 and g(4) = 4^3 = 64. Difference = 81 - 64 = 17. Not more than 1000.
   Full step-by-step solution

   Step 1: Calculate f(4) = 3^4 = 81 and g(4) = 4^3 = 64. Difference = 81 - 64 = 17. Not more than 1000. Step 2: Calculate f(5) = 3^5 = 243 and g(5) = 5^3 = 125. Difference = 243 - 125 = 118. Not more than 1000. Step 3: Calculate f(6) = 3^6 = 729 and g(6) = 6^3 = 216. Difference = 729 - 216 = 513. Not more than 1000. Step 4: Calculate f(7) = 3^7 = 2187 and g(7) = 7^3 = 343. Difference = 2187 - 343 = 1844. This is more than 1000. Therefore, the first positive integer x greater than 3 where f(x) exceeds g(x) by more than 1000 is x = 7. The answer is 7.

6. 2^4 × 3^2 - 5^2 = ? Answer: 119 Solution: Calculate 2^4 = 2 × 2 × 2 × 2 = 16 Calculate 3^2 = 3 × 3 = 9 Calculate 5^2 = 5 × 5 = 25 Multiply 16 × 9 = 144 Subtract 144 - 25 = 119 The answer is 119.
   Full step-by-step solution

   Step 1: Calculate 2^4 = 2 × 2 × 2 × 2 = 16 Step 2: Calculate 3^2 = 3 × 3 = 9 Step 3: Calculate 5^2 = 5 × 5 = 25 Step 4: Multiply 16 × 9 = 144 Step 5: Subtract 144 - 25 = 119 The answer is 119.

7. 2^4 × 3^2 ÷ 6 = ? Answer: 24 Solution: Calculate the exponents first: 2^4 = 16 and 3^2 = 9 Multiply the results: 16 × 9 = 144 Divide by 6: 144 ÷ 6 = 24 The answer is 24.
   Full step-by-step solution

   Step 1: Calculate the exponents first: 2^4 = 16 and 3^2 = 9 Step 2: Multiply the results: 16 × 9 = 144 Step 3: Divide by 6: 144 ÷ 6 = 24 The answer is 24.

8. Compare f(x) = 14^x and g(x) = x^14. At what integer value of x > 13 does f(x) first exceed g(x)? Answer: 15 Solution: Test x = 14 f(14) = 14^14 = 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 = 111,120,068,255,580 (approximately 1.1112 × 10^14) g(14) = 14^14 = 111,120,068,255,580 (same value, so they are equal) Test x = 15 f(15) = 14^15 = 14 × 14^14 = 14 × 111,120,068,255,580 =…
   Full step-by-step solution

   Step 1: Test x = 14 f(14) = 14^14 = 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 × 14 = 111,120,068,255,580 (approximately 1.1112 × 10^14) g(14) = 14^14 = 111,120,068,255,580 (same value, so they are equal) Step 2: Test x = 15 f(15) = 14^15 = 14 × 14^14 = 14 × 111,120,068,255,580 = 1,555,680,955,578,120 (approximately 1.5557 × 10^15) g(15) = 15^14 = 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 × 15 = 291,929,260,253,906 (approximately 2.9193 × 10^14) At x = 15, f(15) = 1,555,680,955,578,120 and g(15) = 291,929,260,253,906. Since 1,555,680,955,578,120 > 291,929,260,253,906, f(x) exceeds g(x) at x = 15. Step 3: Since f(14) = g(14) and f(15) > g(15), the first integer value greater than 13 where the exponential function exceeds the polynomial function is x = 15. The answer is 15.